The Paradox of Knowability from an Intuitionistic Standpoint

Abstract An intuitionistic solution to the Paradox of Knowability is given. It consists (i) in accepting α → K α, the ordinary formalization of the principle of Radical Anti-Realism (RAR) that “Every truth is known”, since, intuitionistically understood, it means that proofs are epistemically transparent; and (ii) in accepting (RAR) itself, on the basis of the fact that knowledge is an intuitionistic internal truth notion. Some neo-verificationist approaches are criticized. Finally the problem of how to frame a rational discussion between Classicism and Intuitionism is briefly discussed.

Introduction

The Paradox of Knowability [1] is an argument that from the principle of Knowability

(K) Every truth is knowable, accepted by anti-realists of all sorts, derives the principle of Radical Anti-Realism (RAR) Every truth is known,

which, on the contrary, virtually no anti-realist would be prepared to accept. Contraposing, if one does not want to accept (RAR), one must reject (K) as well [2].

In a nutshell, the argument has the following structure. I shall use the symbols

&, +, ⊃, =, −, IT, 2: for the classical logical constants; and ∧, ∨, →, ↔, ¬, ∀, ∃ for the intuitionistic ones. First, (K) and (RAR) are formalized by the two following schemas, respectively:3

(1)

(2) α ⊃ ♦ K α α ⊃ K α.

Then replace α in (1) with the proposition “q & − K q”; you obtain the following instance:

(3) (q & − K q) ⊃ ♦ K(q & − K q),

from which it is not difficult to derive, by means of intuitively acceptable principles, the unacceptable (2). The principles are the following:

(a)

(b)

(c)

(d)

and this is the derivation:4 D(K(α & β) ⊃ (K α &K β)

D(K α ⊃ α)

D((α & −α) ⊃ ⊥)

−(α & −β) ⊃ (α ⊃ β),

[q & − K q]1 by (3)

♦K (q & −K q) by (a)

♦(K q &K−K q) by (b)

♦(K q & −K q) by (c)

♦(⊥) −♦(⊥)

⊃E

⊥ −I, 1

−(q & −K q) by (d)

q ⊃ K q

The paradox is usually viewed as an argument against anti-realism, when this is conceived, according to a famous proposal by Michael Dummett, as a doctrine concerning meaning rather than ontology. Dummett writes:

Realism I characterise as the belief that statements of [a certain] class possess an objective truth-value, independently of our means of knowing it: they are true or false in virtue of a reality existing independently of us. The anti-realist opposes to this the view that the statements of [that] class are to be understood only by reference to the sort of thing which we count as evidence for a statement of that class.5

The fundamental opposition between realism and anti-realism concerns therefore, according to Dummett, the key notion of the theory of meaning, i.e. the notion in terms of which the meaning of the statements of the given class is to be explained: truth according to the realist, evidence according to the anti-realist. However, since Dummett holds that meaning is to be explained in any case in terms of truth-conditions, and that for the anti-realist truth can consist only in the existence of evidence, the realism/anti-realism opposition, in the final version he offers, concerns the notion of truth to be used in explaining meaning: the bivalent notion according to the realist, some non-bivalent notion according to the anti-realist.6 The criterion of realism is therefore, in Dummett's opinion, the acceptance/refusal of the bivalence principle concerning truth.

Notice that this raises immediately a question: if we are ready to oppose the realistic, bivalent, notion of truth with other, non-bivalent, notions, we must ask at which conditions a notion can be considered as a notion of truth. I shall return to this question later. For the time being let me observe that the story of the semantic characterization of the realism/anti-realism debate is not finished. After Dummett it has been observed that all the anti-realistic notions of truth on the market (truth as assertibility, truth as existence of a verification, and so on) share a general characteristic: that truth is an epistemic notion, and therefore is essentially knowable. Knowability has therefore been identified as the essential property of truth, and anti-realism has been characterized as the view that every truth is knowable. It is precisely this feature of anti-realistic truth that the paradox is intended to hit.

In this paper I shall first argue that an intuitionistic solution to the paradox is available; then I shall examine the position of neo-verificationism concerning the paradox; finally I shall briefly consider the general question of how a rational discussion of alternative logics is possible at all.

[1] The paradox is usually ascribed to F. Fitch, but is due in fact to A. Church. For its history see [18]

[2] Of course, an intuitionist might also accept the negation of (K) without accepting that there are unknowable truths